Solvability of cubic and quartic equations using one radical

TL;DR

This paper characterizes when cubic and quartic polynomials with rational coefficients can be solved using a single radical extension, providing explicit conditions involving discriminants and rational roots.

Contribution

It offers new necessary and sufficient conditions for solving certain cubic and quartic equations with one radical extension, extending classical solvability criteria.

Findings

Cubic polynomial root solvability depends on discriminant being a rational square.

Quartic polynomial root solvability depends on a rational root of the cubic resolution and a specific rational square condition.

Provides explicit algebraic criteria for one radical solvability of cubic and quartic equations.

Abstract

Theorem. An irreducible cubic polynomial with rational coefficients has a root in a one step radical extension of Q if and only if the discriminate is a square of a rational number. Theorem. An irreducible polynomial x^4+px^2+qx+s with rational coefficients q\ne0, p and s has a root in a one step radical extension of Q if and only if the cubic resolution has rational root t such that t>p/2 and A:=16(t^2-s)^2-(t^2-s)(2t+p)^2 is a square of a rational number.